availability: Availability Function
In smmR: Simulation, Estimation and Reliability of Semi-Markov Models

Description Usage Arguments Details Value References

The pointwise (or instantaneous) availability of a system S_{ystem} at time k \in N is the probability that the system is operational at time k (independently of the fact that the system has failed or not in [0, k)).

1	availability(x, k, upstates = x$states, level = 0.95, klim = 10000)

`x`	An object of S3 class `smmfit` or `smm`.
`k`	A positive integer giving the time at which the availability should be computed.
`upstates`	Vector giving the subset of operational states U.
`level`	Confidence level of the asymptotic confidence interval. Helpful for an object `x` of class `smmfit`.
`klim`	Optional. The time horizon used to approximate the series in the computation of the mean sojourn times vector m (cf. meanSojournTimes function) for the asymptotic variance.

Consider a system (or a component) S_{ystem} whose possible states during its evolution in time are E = \{1,…,s\}. Denote by U = \{1,…,s_1\} the subset of operational states of the system (the up states) and by D = \{s_1 + 1,…,s\} the subset of failure states (the down states), with 0 < s_1 < s (obviously, E = U \cup D and U \cap D = \emptyset, U \neq \emptyset,\ D \neq \emptyset). One can think of the states of U as different operating modes or performance levels of the system, whereas the states of D can be seen as failures of the systems with different modes.

We are interested in investigating the availability of a discrete-time semi-Markov system S_{ystem}. Consequently, we suppose that the evolution in time of the system is governed by an E-state space semi-Markov chain (Z_k)_{k \in N}. The state of the system is given at each instant k \in N by Z_k: the event \{Z_k = i\}, for a certain i \in U, means that the system S_{ystem} is in operating mode i at time k, whereas \{Z_k = j\}, for a certain j \in D, means that the system is not operational at time k due to the mode of failure j or that the system is under the repairing mode j.

Thus, the pointwise availability of a semi-Markov system at time k \in N is

A(k) = P(Z_k \in U) = ∑_{i \in E} α_i A_i(k),

where we have denoted by A_i(k) the conditional availability of the system at time k \in N, given that it starts in state i \in E,

A_i(k) = P(Z_k \in U | Z_0 = i).

A matrix with k + 1 rows, and with columns giving values of the availability, variances, lower and upper asymptotic confidence limits (if x is an object of class smmfit).

V. S. Barbu, N. Limnios. (2008). Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications - Their Use in Reliability and DNA Analysis. New York: Lecture Notes in Statistics, vol. 191, Springer.

smmR documentation built on Aug. 3, 2021, 5:07 p.m.